MinT - Best Known (229−90, 229, s)-Nets in Base 4

Best Known (229−90, 229, s)-Nets in Base 4

(229−90, 229, 138)-Net over F₄ — Constructive and digital

Digital (139, 229, 138)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (21, 66, 34)-net over F₄, using
  - net from sequence [i] based on digital (21, 33)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 21 and N(F) ≥ 34, using
      - T₅ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]
2. digital (73, 163, 104)-net over F₄, using
  - net from sequence [i] based on digital (73, 103)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 104, using
      - F₆ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(229−90, 229, 349)-Net over F₄ — Digital

Digital (139, 229, 349)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(229−90, 229, 6768)-Net in Base 4 — Upper bound on s

There is no (139, 229, 6769)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 745695 533795 376990 305773 369264 636277 941627 605388 942861 182432 620875 021632 205283 841349 575064 997362 969722 566855 596148 335623 464450 618391 236416 > 4²²⁹ [i]